Lesson 5

Construction Techniques 3: Perpendicular Lines and Angle Bisectors

  • Let’s use tools to solve some construction challenges.

5.1: Two Circles

Points \(A\) and \(B\) are each at the centers of circles of radius \(AB\).

Two circles centered at A and B, each pass through the center of the other and intersect at E and F. Radius AB forms a horizontal line segment.
  1. Compare the distance \(EA\) to the distance \(EB\). Be prepared to explain your reasoning.
  2. Compare the distance \(FA\) to the distance \(FB\). Be prepared to explain your reasoning.
  3. Draw line \(EF\) and write a conjecture about its relationship with segment \(AB\).

5.2: Make It Right

Here is a line \(\ell\) with a point labeled \(C\):

Use straightedge and compass tools to construct a line perpendicular to \(\ell\) that goes through \(C\).

5.3: Bisect This

Here is an angle:

 
  1. Estimate the location of a point \(D\) so that angle \(ABD\) is approximately congruent to angle \(CBD\).
  2. Use compass and straightedge tools to create a ray that divides angle \(CBA\) into 2 congruent angles. How close is the ray to going through your point \(D\)?
  3. Take turns with your partner, drawing and bisecting other angles.

    1. For each angle that you draw, explain to your partner how each straightedge and compass move helps you to bisect it.

    2. For each angle that your partner draws, listen carefully to their explanation. If you disagree, discuss your thinking and work to reach an agreement.



For thousands of years since the ancient Greeks started playing with straightedge and compass constructions, people strived to find a construction to trisect an arbitrary angle into three equal angles. Many claimed to have found such a construction, but there was always some flaw in their reasoning. Finally, in 1837, Pierre Wantzel used a new field of mathematics to prove it was impossible—which still did not stop some from claiming to have found a construction. If we allow other tools besides just a straightedge and compass, though, it is possible. For example, try this method of using origami (paper folding) to trisect an angle. 

Video Trisecting an Angle with Origami available at https://player.vimeo.com/video/298418799.

 

Image of unfolded origami paper.

 

Summary

We can construct a line that is perpendicular to a given line. We can also bisect a given angle using only a straightedge and compass. The line that bisects an angle is called the angle bisector. Both constructions use 2 circles that go through each other’s centers:

Two identical circles each passes through the center of the other.

For the perpendicular line, start by finding 2 points on the line the same distance from the given point. Then create the 2 circles that go through each other’s centers. Connect the intersection points of those circles to draw a perpendicular line.

Two circles that go through each other's centers.

For the angle bisector, start by finding 2 points on the rays the same distance from the vertex. Then create the 2 circles that go through each other’s centers. Connect the intersection points of those circles to draw the angle bisector.

A construction of angle bisector with circles.

In fact, we can think of creating a perpendicular line as bisecting a 180 degree angle!

Glossary Entries

  • angle bisector

    A line through the vertex of an angle that divides it into two equal angles.

  • circle

    A circle of radius \(r\) with center \(O\) is the set of all points that are a distance \(r\) units from \(O\). 

    To draw a circle of radius 3 and center \(O\), use a compass to draw all the points at a distance 3 from \(O\).

  • conjecture

    A reasonable guess that you are trying to either prove or disprove.

  • inscribed

    We say a polygon is inscribed in a circle if it fits inside the circle and every vertex of the polygon is on the circle. We say a circle is inscribed in a polygon if it fits inside the polygon and every side of the polygon is tangent to the circle.

     

  • line segment

    A set of points on a line with two endpoints.

  • parallel

    Two lines that don't intersect are called parallel. We can also call segments parallel if they extend into parallel lines.

  • perpendicular bisector

    The perpendicular bisector of a segment is a line through the midpoint of the segment that is perpendicular to it.